Abdullah Harmanci

On ⌖-supplemented Modules

Let R be a ring and M a right R-module. M is called ⌖-supplemented if every submodule of M has a supplement that is a direct summand of M, and M is called completely ⌖-supplemented if every direct summand of M is ⌖-supplemented. In this paper various properties of these modules are developed. It is shown that (1) Any finite direct sum of ⌖-supplemented modules is ⌖-supplemented. (2) If M is ⌖-supplemented and (D3) then M is completely ⌖-supplemented.

On a Class of Semicommutative Rings

In this paper, a generalization of the class of semicommutative rings is inves- tigated. A ring R is called central semicommutative if for any a;b 2 R, ab = 0 implies arb is a central element of R for each r 2 R. We prove that some results on semicommutative rings can be extended to central semicommutative rings for this general settings.

On prime submodules and primary decomposition

We characterize prime submodules of R × R for a principal ideal domain R and investigate the primary decomposition of any submodule into primary submodules of R × R.

On the Pure-injectivity Profile of a Ring

An analog of the injective profile of a ring, with relative injectivity replaced by relative pure-injectivity, is investigated. Emphasis is placed on comparing and contrasting the properties of both profiles. In particular, the analog in this context of the notion of poor modules is considered and properties of pure-injectively poor modules are determined. While we do not know of any ring that does not have pure-injectively poor modules, their existence has not been determined in general. Rings having pure-injectively poor modules of various types are characterized.

Modules in Which Inverse Images of Some Submodules are Direct Summands

Let R be an arbitrary ring with identity and M a right R-module with S = EndR(M). Let F be a fully invariant submodule of M. We call M an F-inverse split module if f−1(F) is a direct summand of M for every f ∈ S. This work is devoted to investigation of various properties and characterizations of an F-inverse split module M and to show, among others, the following results: (1) the module M is F-inverse split if and only if M = F ⊕ K where K is a Rickart module; (2) for every free R-module M, there exists a fully invariant submodule F of M such that M is F-inverse split if and only if for every projective R-module M, there exists a fully invariant submodule F of M such that M is F-inverse split; and (3) Every R-module M is Z2(M)-inverse split and Z2(M) is projective if and only if R is semisimple.

Modules Which Are Lifting Relative To Module Classes

In this paper, we study a module which is lifting and supplemented relative to a module class. Let R be a ring, and let X be a class of R-modules. We will define X-lifting modules and X-supplemented modules. Several properties of these modules are proved. We also obtain results for the case of specific classes of modules.

Rickart modules relative to singular submodule and dual Goldie torsion theory

Let R be an arbitrary ring with identity and M a right R-module with the ring S = EndR(M) of endomorphisms of M. The notion of an F-inverse split module M, where F is a fully invariant submodule of M, is defined and studied by the present authors. This concept produces Rickart submodules of modules in the sense of Lee, Rizvi and Roman. In this paper, we consider the submodule F of M as Z(M) and Z∗(M), and investigate some properties of Z(M)-inverse split modules and Z∗(M)-inverse split modules M. Results are applied to characterize rings R for which every free (projective) right R-module M is F-inverse split for the preradicals such as Z(⋅) and Z∗(⋅).

Feckly reduced rings

Let R be a ring with identity and J(R) denote the Jacobson radical of R. In this paper, we introduce a new class of rings called feckly reduced rings. The ring R is called feckly reduced if R=J(R) is a reduced ring. We investigate relations between feckly reduced rings and other classes of rings. We obtain some characterizations of being a feckly reduced ring. It is proved that a ring R is feckly reduced if and only if every cyclic projective R-module has a feckly reduced endomorphism ring. Among others we show that every left Artinian ring is feckly reduced if and only if it is 2-primal, R is feckly reduced if and only if T(R;R) is feckly reduced if and only if R[x]= is feckly reduced.

Strongly Large Module Extensions

Abstract In this paper, we introduce strongly large submodules and investigate their properties. A submodule N of a right R-module M is said to be strongly large in case for any m ∈ M, s ∈ R with ms ̸= 0 there exists an r ∈ R such that mr ∈ N and mrs ̸= 0. In this note, we also define and study strongly large closed submodules and strongly large complement submodules

A dual approach to the theory of inverse split modules

Let R be an arbitrary ring with identity, M a right R-module and F a fully invariant submodule of M. The notion of an F-inverse split module M has been defined and studied by the present authors recently. In this paper, we introduce its dual notion, namely, dualF-inverse split moduleM . This work is devoted to investigation of various properties and characterizations of a dual F-inverse split module M. We include applications for rings and cosingular submodules. We also deal with the notion of relatively dual inverse splitness to investigate direct sums of dual inverse split modules.